Quadratica
by Neutrina
Summary: I wrote this for math class, so it's mostly about the math. I know it's a little confusing.


It was a dark night. Catherine Willows of the Las Vegas Crime Lab was having coffee with Grissom when she got the call: a mother frantically looking for her daughter in California. Because it was connected to a series of kidnappings in Vegas, Catherine and Grissom left for Mountain View. After an excruciating flight on a cramped helicopter with some bugs (which Grissom declared were _vegrandis vexo susurro res,) _they arrived at the lab. It was nothing like the lab in Vegas. For one thing, it was located behind a church. More importantly, it was a school during the day. The Girls' Middle School, to be exact.

Catherine and Grissom walked to the staff lounge (the California version of the break room, plus better food) for a debriefing. Apparently, the girl was in a locked room, and the window, which only opens from the inside, was locked as well. The fact that there was no sign of a struggle would have signified a run away, but there was a note taped to the door: x²+5x+4

Both Catherine and Grissom were stumped at first. This changed when the trace expert said, "We were working on factor books in class when she was abducted. Because she knew about the lab, it's reasonable to assume it's a note to us."

Grissom looked thoughtful. "What all can we do with this?"

Again, the trace expert was happy to help. "This can be factored in a variety of ways. One of them could lead to an address. Because this is an expression, and therefore doesn't have an equals sign, it cannot be graphed or solved. It can, however, be factored." She drew an x on a handy piece of graph paper.

"This," she explained, "is called a diamond problem. The product goes in the top section, the sum goes in the bottom section, and the numbers that follow this rule go on the sides. In the case of an expression like this, the product is the purely number term, called the _constant_, in this case 4. The sum is the coefficient of x, which in this case is 5. If there is no x term or constant, the value for that term is 0. Got that?" both Grissom and Catherine nodded. Catherine took the pencil from her, and began to add to he drawing.

"So what two numbers can we add to make 5, and multiply to make 4?" asked Catherine.

Grissom responded, "What are the factors of 4?" Evidently this was a rhetorical question, as he answered himself. "2,2 and 4,1. Two plus two is four, so the only factors we're left with are one and four. One plus four is five." He added this to the piece of paper.

The trace expert, whose name was revealed to be Esther, nodded approvingly. "Now you write it." She writes x²+5x+4(x+4)(x+1)

Grissom's eyes lit up.

"If we write (x+4)(x+1)0, then we can graph it. Maybe it will look like a map!" He began to solve for the roots.

**(x+4)(x+1)0**

**x-4, -1**

"Why did you do that?" asked Catherine, looking confused. "Why will that help?" Grissom, as always, decided to take the long way around.

"What I just did is called solving for the roots of a parabola. The roots are the x intercepts, and there can be 0,1,or 2 roots depending on where the vertex is. In this case, the vertex is below the x-axis, so there are two roots. I found them by using the Zero Product Property, which is that if abc0 then either a0, b0, c0, or some combination. The Zero Product Property is used to solve for x when an equation is equal to zero, and it helps find the roots of the parabola, which I need to do to graph this. Because 4+-40,x can equal –4, and because 1+-10, x can equal –1." Catherine only looked more confused.

"Why zero?"

"The x-intercepts are where y equals zero. I substituted y with 0 in the equation (x+4)(x+1)y to find the roots."

"Ok, now you tell me in English."

"To get x²+5x+4, I multiplied (x+4) by (x+1). So, to get (x+4)(x+1), I undid the multiplication. Now comes the fun part."

Catherine rolled her eyes. "And what would that be, pray tell?"

Grissom smiled. "Graphing." On the handy piece of graph paper, he began to draw.

"I know that the vertex is between the two roots, which is why the middle point is green; it's marking the x-value of the vertex

Because I have the x-value, I can find the y-value with substitution. I know that the x-value is –2.5, so I substitute x with –2.5 in x²+5x+4y, and get (-2.5) ²+5(-2.5)+4y. Now I can use the order of operations to solve for y." Grissom quickly solved it on the now close to full piece of graph paper.

6.25-12.5+4y

2.25y

"Now I know that the vertex is (-2.5, -2.25). With the vertex and the roots, I can graph the parabola." He did just that, adding once more to the beautiful piece of graph paper.

The group seemed to be at a dead end when another one of the local CSIs gasped in recognition. "That looks like The Shipwreck!" At the other's inquisitive stares, she clarified. "It's a restaurant. Not that many people have heard about it, though. It's in the middle of nowhere, but it serves good piña coladas."

Before she could finish speaking, the others were out the door, and halfway to the parking lot. She shrugged apologetically at Catherine and Grissom. "They're just excited to have a lead."

Catherine raised her eyebrows. "Remind you of someone?" Grissom laughed.

The Shipwreck was closed for remodeling. Why you would need to remodel a sunken ship, Catherine didn't know. There was another note taped to the door: (x+3)(x+5)

It was Catherine's turn for inspiration. "Are there any ways to multiply this that will require some kind of diagram or graph?"

Esther began to explain a curious thing called a Generic Rectangle. "Generic Rectangles are something invented by CPM© to a polynomial by another polynomial. There are three common types of polynomials. A monomial is an expression with one term, such as 4, or x, a binomial is an expression with two terms like (x+4), and a trinomial is an expression with three terms, such as x+y+4, or, more commonly, x²+5x+6. And just so you know what the heck we're dealing with, this is a quadratic equation. A quadratic equation is an equation in which there is an x², but no higher exponents. The standard form for a quadratic equation is ax²+bx+c. Got it? Ok, moving on. In this case, we use a 2 x 2 rectangle." She began drawing.

x + 3

x

5

"Now we fill it out."

x + 3

x x² 3x

5 5x 15

"Ok, now we just have to figure out what to do with this."

Yet another CSI's eyes lit up. "If we use only the coefficients, and make them apartment numbers, that looks just like this weird apartment complex near my house!" Again, everyone rushed to the parking lot.

There were four buildings; Building 1, Building 3, Building 5, and Building 15. "Great, what apartment do we check?" asked a CSI. Catherine, enjoying the math, came up with the answer. "Let's write out the equation."

(x+5)(x+3)x²+8x+15

"So," she concluded, "Let's try apartments 1, 8, and 15."

At apartment 1, there was a note saying (x+3)². Catherine looked puzzled. "That's what Lindsay was doing at school!" she said with a sudden burst of recognition. "Perfect square binomials! I remember she was really excited because she found a pattern." She looked deep in thought for a minute. "I remember! It was that expressing it as a sum was x²+ 2x(the constant) +(the constant)²! So if it's (x+6)2, the answer would be x2 +12x+36! And if it's (x-6)2, the answer is x2 –6x+36. So in this case, our answer is x²+6x+9!"

Everybody trooped over to apartment 6, where there was yet another note on the door; (x+2)(x²+x+1) "Can we use a 2 x _3 _Generic Rectangle for this?" asked Catherine excitedly. Instead of answering, Esther drew a 2 x 3 Generic Rectangle.

x² + x + 1

x x³ x² x

2x² 2x 2

2

"So our answer is x³+ 3x² +3x+2."

"Hey, how do you do a Generic Rectangle with negatives?" asked Catherine excitedly.

x - 5

x x2 -5x

6 6x -30 x2+6x-5x-30x2+x-30

Catherine nodded in comprehension. Everyone headed over to apartment 3, where there was yet another note: 3x² +5x+2. Without even needing to be asked, Esther started explaining. "When the coefficient of x is not equal to one, you need to check for a common factor. For example, take 3x2+6x+9. The common factor is 3. Then you write 3(x2+2x+3), and factor from there. 3, 5, and 2 have no common factor other than one, which won't help. If there isn't one, there is a special process. First, you make a diamond problem. In the top you put the product of the coefficient of x² and the constant. In the bottom, you put the coefficient of x. Then you solve it."

6

2 3

5

"You use these to make a generic rectangle. The sides of the diamond problem are the coefficients of x inside the rectangle."

x + 1

3x 3x² 3x

2 2x 2

"Now we can write 3x² +5x+2(x+1)(3x+2). "

Because they had already been to apartment one, they went to apartment 2. Finally, there was a note that was something other than math. Unfortunately, it was in German. It said something along the lines of "Hilfe! wir ziehen auf die goldene Gatterbrücke um!". Fortunately, one of the CSIs brought her laptop. They found an online translator, and translated the note. It took a while, on account of the computer running on Windows 95, but finally they figured out what the note said: "Help! We're moving to the Golden Gate Bridge!" They drove to the Golden Gate, but all there was was another note. Everyone considered himself or herself lucky to find it, as it was on a piece of newspaper on the sidewalk. It said (x-4)(x+4)password. Location is first 3 Fibonacci numbers on Gold (the color, not the metal) in Spanish Street.

Catherine was the first to speak. "We need to do this quickly. Are there any short-cuts when the constants are the same?" Esther was pleased that Catherine had caught on so quickly.

"As a matter of fact, yes. For any equation that is the difference of two squares, which is the special mathematical term (don't you feel special now?), all you have to do is square the coefficient of x, and have that as the coefficient of x², and subtract from that the constant squared. I know that made no sense, so I'll show you." Grissom handed over the graph paper.

(x+4)(x-4) x²-16

"The coefficient of x is 1, and 1 squared is 1. 4 squared is 16. So, expressing (x+4)(x-4) as a sum is 1x²-16. A simpler way to write this is just x²-16. I'm going to show you why this works. Oh, and gold in Spanish is dorado."

She began writing on the ever so lovely piece of graph paper. "I'm going to start with an x by x square."

x

"Now, because 4 times –4 is negative, I'm going to take out a 4 by 4 square."

x

x

"I can move the bottom square up to the top so I have a rectangle."

x - 4

x + 4

"This rectangle is (x-4)(4+4). The area is x2-4x+4x-16. The x's add to 0, so the area is x²-16" Catherine had loved Fibonacci numbers, so she still remembered the first few: 1,1,2,3,5,8,13,21,34,55,81… Everyone drove as fast as they could, while not getting pulled over for speeding, to 112 Dorado Street. As they expected, there was a keypad on the door. Oddly enough, there was a negative sign as one of the digits. Esther typed in -16 with a shaking hand, and the door slowly swung open. Being bright CSIs, they actually had a warrant and guns, so the group slowly slunk in. Scarily enough, the inside looked like a school. There was a note hastily scrawled on the wall, saying room (x+2)(x+3). Password x²+3x+5. Nearest hundredth. Desperately, Catherine screeched "I don't have time for this! Is there any way we can solve this without an annoying little diagram?" She noticed everyone staring at her. "Well, is there?" she demanded.

"Yes, already!" growled Esther. "It's called the F.O.I.L method."

Catherine scoffed. "Do you really think I'm going to believe that we need to use aluminum to solve this thing?" Esther rolled her eyes venomously.

"F.O.I.L stands for First Outside Inside Last." She tore the graph paper out of her jacket pocket.

She drew (x+2)(x+3)on the battered piece of paper.

"First, multiply the first terms, x and x."

x²

"Now, multiply the inside terms, 2 and x"

2x

"Ok, now the outside terms, x and 3."

3x

"Lastly, multiply the last terms, 2 and 3."

6

"So, our answer is x²+5x+6. Let's check room 6." Exclaimed Catherine.

When they got there, there was another keypad on the wall. "Okay," said Grissom slowly. "How do we solve x²+3x+1? I can't figure out how the heck to factor it."

"Sit down." Sighed Esther. "This is going to take a while."

"Once, when the world was still young, one young goddess created math. Then, her rival created the unfactorable quadratic equation so people would not like math. To counter this, the young goddess, named Quadratica, created the glorious Quadratic Formula, so the people could solve for x in peace. Or so I told my students. I doubt they believed me, but I really don't care. The Quadratic Formula goes like this: " She started singing a song to the tune of Jingle Bells

"The Opposite of b

Plus or minus radical

b squared minus four a c

All divided by two

a!" Everyone applauded. "Great, now I'm going to show you what that meant." She pulled out a shiny new piece of graph paper, and drew:

x-b b²-4ac

2a

"Now, I'm going to solve this. First, I'm going to define a, b, and c. a is the coefficient of x squared, b is the coefficient of x, and c is the constant." Esther quickly jotted a1, b3, c1 on the stunning piece of graph paper. Everyone quickly oohed and ahhed at the new pencil on the new paper, then craned their necks to see better.

"Now, I'm going to substitute in the a, b, and c values into the equation."

x-3 3²-4(1)(1)

2(1)

"Now I need to simplify it to find exact solutions."

x-3 9-4

2

x-3 5

2

"To get exact solutions, I leave in the square root sign."

x-3+ 5 or x-3- 5

2 2

"Finally, I can approximate the values of x. And, in case you're wondering, you **_DO_** have to do all six steps, and you **_DO_** have to write 'or'. If you write 'and', it **_WILL_** be incorrect." She glared at the assembled group, then continued writing.

x  -0.38,-2.62

"These are the x values in the x intercepts in a parabola, also called the roots. It is standard to round to the nearest hundredth. So now we have two passwords, assuming we aren't mixing the digits up: 038262, or 262038."

"26 was my lucky number in middle school," interjected Catherine. "Let's use the one that starts with that first." She swiftly punched in the combination, and the door swung open. There were two sets of muddy footprints leading to a section in the wall. Grissom pressed it, and the wall moved to reveal a well-lit hallway.

"Ladies first," smirked Grissom. Snarling, Catherine pushed into the passage, closely followed by Esther. The window at the end was broken. Grissom opened it, and the group crawled through the vent that was revealed. They found themselves in a small attic-like space.

There was a girl tied to the wall. She had high-tech handcuffs on, though, with several passwords. The kidnapper put hints on, thinking that no one would get the math-related jargon, but this kidnapper didn't know whom s/he was dealing with. The first word hint popped up on the minuscule screen. "The value under the square root sign in a q-equation…" Esther scoffed. "Oh, that's hard to guess." She quickly typed in 'discriminant' in the keypad. At the other's questioning stares, she explained. "In a quadratic equation, the value under the square root sign, and the square root sign is called the discriminant because it allows you to discriminate whether an equation is factorable. If it's not factorable, you have to solve it with the quadratic formula, although if you're looking at the discriminant, you should probably just keep solving, rather than go back and try to factor it. Here, I'll show you while we wait for these."

"If a quadratic equation is: x -552-4(2)(2)

2(2)

you solve the discriminant: 5 squared is 25. 2 times 2 times 4 is 16. So I can write it as x -525-16

4

I can write the discriminant as 9 . The square root of 9 is rational, so the quadratic equation is factorable."

She waited patiently for the next hint to pop up. "A number that goes on forever is…"

Catherine lunged for the keypad, and typed in 'irrational.' "I remember this from middle school! If the discriminant is not a perfect square, the values for x will be irrational! And if the discriminant is a perfect square, or if the equation is factorable, the x-values will be rational!" She took a minute to get her breath, then continued. "A rational number is something that doesn't go on forever, whether it's a fraction, decimal, or integer. Also, if it goes on forever, but is a repeating pattern, like the decimal equivalent of 1/7, it's rational." The keypad beeped softly, and the handcuffs opened. Suddenly, an alarm went off downstairs. Footsteps hurried towards the wall. "Get down!" screamed Catherine.

Everyone who had a gun pulled it out, and the girl ducked.

The wall exploded in a cloud of debris and bullets. Then, just as soon as it began, it was over. The kidnapper was surrounded, and miraculously, no one was dead. The girl finally got home.

When they got back to the school, a golden glow was seen coming from the library. From somewhere in the glow, a beautiful voice sounded. "I am Quadratica! If you were wondering about your improbable luck in always going to the right place, I will put your minds to rest. I have been watching over you!" The glow slowly receded, until there was only an echo. "Never forget the beauty of math…" It faded slowly.

Later, in the interrogation room, Grissom had to ask, even though he knew he would never fully understand. "Why? Why do you abduct children?"

The kidnapper smiled chillingly. "Because I can."


End file.
